Domino tilings of rectangles with fixed width

Let t(k,n) denote the number of ways to tile a 1 x n rectangle with 1 x 2 rectangles (called dominoes). We show that for each fixed k the sequence t"k=(t(k,0), t(k,1),...) satisfies a difference equation (linear, homogeneous, and with constant coefficients). Furthermore, a computational method is given for finding this difference equation together with the initial terms of the sequence. This gives rise to a new way to compute t(k,n) which differs completely with the known Pfaffian method. The generating function of t"k is a rational function F"k, and F"k is given explicitly for k=1,...,8. We end with some conjectures concerning the form of F"k based on our computations.